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Multiple solutions for singular semipositone boundary value problems of fourth-order differential systems with parameters

Abstract

The aim of this paper is to establish some results about the existence of multiple solutions for the following singular semipositone boundary value problem of fourth-order differential systems with parameters:

$$ \textstyle\begin{cases} u^{(4)}(t)+\beta _{1}u''(t)-\alpha _{1}u(t)=f_{1}(t,u(t),v(t)),\quad 0< t< 1; \\ v^{(4)}(t)+\beta _{2}v''(t)-\alpha _{2}v(t)=f_{2}(t,u(t),v(t)),\quad 0< t< 1; \\ u(0)=u(1)=u''(0)=u''(1)=0; \\ v(0)=v(1)=v''(0)=v''(1)=0, \end{cases} $$

where \(f_{1},f_{2}\in C[(0,1)\times \mathbb{R}^{+}_{0}\times \mathbb{R}, \mathbb{R}]\), \(\mathbb{R}_{0}^{+}=(0,+\infty )\). By constructing a special cone and applying fixed point index theory, some new existence results of multiple solutions for the considered system are obtained under some suitable assumptions. Finally, an example is worked out to illustrate the main results.

Introduction

In the recent decades, the topic about the existence of solutions of nonlinear boundary value problems (BVPs for short) has received considerable popularity due to its wide applications in biology, hydrodynamics, physics, chemistry, control theory, and so forth. Some progress has also been made in the study of solutions for various types of equations or systems including differential equation [13, 21, 25, 27], integro-differential equation [2, 19, 27], evolution equations [1, 7], fractional systems [3, 15, 17, 2224, 30, 31], impulsive systems [14, 18, 28], and delay systems [14]. In consequence, many meaningful results have been obtained in these fields. For more details, please see Lakshmikantham et al. [8], Podlubny [16], and the references therein.

As a branch of research on boundary value problems, singular boundary value problems arise from many fields, such as nuclear physics, biomathematics, mechanics or engineering, and play an extremely important role in both theoretical developments and practical applications [5, 1012, 26, 29, 32]. Moreover, extensive attention has been drawn to the study of singular semipositone boundary value problems (SBVPs for short) to differential equations or systems recently. For example, in [12] Y. Liu investigated the existence of two positive solutions to the singular semipositone problem

$$ \textstyle\begin{cases} y^{\prime \prime }+\lambda f(t,y)=0, \quad 0< t< 1; \\ y(0)=y(1)=0, \end{cases} $$

where \(f\in C[J\times \mathbb{R}_{0}^{+},\mathbb{R}]\), \(J=(0,1)\), \(\mathbb{R}_{0}^{+}=(0,+\infty )\), and the parameter \(\lambda >0\). The nonlinear term f may be singular at \(t=0\), \(t=1\) and \(y=0\). By constructing a special cone, the existence of multiple positive solutions was obtained under some suitable assumptions. In [32], Zhu et al. considered the existence of positive solutions of the two-point boundary value problem for nonlinear singular semipositone systems

$$ \textstyle\begin{cases} x^{(4)}=f(t,x(t),y(t),x^{\prime \prime }(t),y^{\prime \prime }(t)),\quad 0< t< 1; \\ y^{(4)}=g(t,x(t),y(t),x^{\prime \prime }(t),y^{\prime \prime }(t)),\quad 0< t< 1; \\ x(0)=x(1)=x^{\prime \prime }(0)=x^{\prime \prime }(1)=0; \\ y(0)=y(1)=y{''}(0)=y{''}(1)=0, \end{cases} $$

where \(f,g\in C[J\times \mathbb{R}^{+}\times \mathbb{R}^{+}\times \mathbb{R}^{-}\times \mathbb{R}^{-},\mathbb{R}]\) may be singular at \(t=0\) or \(t=1\), not singular at \(u=0\), \(\mathbb{R}^{+}=[0,+\infty )\), \(\mathbb{R}^{-}=(-\infty ,0]\). By applying the fixed point theory in cones, the existence results of positive solutions were established.

As we all know, fourth-order boundary value problems have important practical applications in physics and engineering, and, for instance, they are usually used to describe the deformation of an elastic beam supported at the end points [4, 9, 20]. Wang et al. [20] investigated the boundary value problems of a class of fourth-order differential systems with parameters as follows:

$$ \textstyle\begin{cases} u^{(4)}(t)+\beta _{1}u''(t)-\alpha _{1}u(t)=f_{1}(t,u(t),v(t)),\quad 0< t< 1; \\ v^{(4)}(t)+\beta _{2}v''(t)-\alpha _{2}v(t)=f_{2}(t,u(t),v(t)),\quad 0< t< 1; \\ u(0)=u(1)=u''(0)=u''(1)=0; \\ v(0)=v(1)=v''(0)=v''(1)=0, \end{cases} $$
(1.1)

where \(f_{1},f_{2}\in C[[0,1]\times \mathbb{R}^{+}\times \mathbb{R}^{+}, \mathbb{R}^{+}]\), and \(\beta _{i},\alpha _{i}\in \mathbb{R}\) (\(i=1,2\)) satisfying

$$ \beta _{i}< 2\pi ^{2},\qquad -\beta _{i}/4\leq \alpha _{i},\qquad \alpha _{i}/ \pi ^{4} +\beta _{i}/\pi ^{2} < 1. $$
(1.2)

The existence results of positive solutions were proved by using the fixed point theory under two novel cones being constructed.

Unfortunately, the result obtained in [20] is only the existence of at least one nontrivial positive solution when the nonlinear terms have no singularity. It should be stressed also in [20] that the solutions of BVPs (1.1) are all positive and the nonlinear terms must be nonnegative, which is limited to a certain extent in some cases. Besides, we know that there is always some connection between the nonlinear terms in practical applications, but the description of this connection is rarely mentioned and studied in the present literature. To our best knowledge, there is no paper considering SBVPs (1.1) when \(f_{1}(t,u,v)\) and \(f_{2}(t,u,v)\) are singular at \(t=0\), \(t=1\), and \(u=0\), and also no result is available about the existence of multiple solutions for such boundary value problems.

Motivated by all the above analyses, in this paper we discuss the existence and multiplicity of solutions to SBVPs (1.1) when the parameters \(\beta _{i},\alpha _{i}\in \mathbb{R}\) (\(i=1,2\)) satisfy condition (1.2). In addition, \(f_{1}, f_{2}\in C[(0,1)\times \mathbb{R}_{0}^{+}\times \mathbb{R}, \mathbb{R}]\), namely \(f_{1}(t,u,v)\) and \(f_{2}(t,u,v)\) may be singular at \(t=0\), \(t=1\) and \(u=0\), and \(f_{1}\), \(f_{2}\) are semipositone rather than positive with some connection imposed between them. Our approaches are based on the approximation method and the well-known fixed point index theory.

Obviously, what we consider is more different from [20] and [32]. The main features of the present work are as follows. Firstly, \(f_{1}(t,u,v)\) and \(f_{2}(t,u,v)\) may be singular at both \(t=0\), \(t=1\) and \(u=0\), and under some suitable assumptions, the multiple nontrivial solutions for SBVPs (1.1) are established. Secondly, \(f_{1}\) may be negative for some values of t, u, and v; \(f_{2}\) is also allowed to change sign. Moreover, \(f_{2}\) is controlled by \(f_{1}\). Thirdly, in the obtained solution \((u,v)\), the component u is positive, but the component v is allowed to have different signs, even may be negative.

The rest of the present work is organized as follows. Section 2 contains some preliminaries. In Sect. 3, some transformations are introduced to convert SBVPs (1.1) into the corresponding approximate boundary value problems. The main results will be given and proved in Sect. 4. Finally, in Sect. 5, an example is given to demonstrate the main result.

Preliminaries

In view of condition (1.2), as in [9], denote

$$ \xi _{i,1}=\frac{-\beta _{i}+\sqrt{\beta _{i}^{2}+4\alpha _{i}}}{2}, \qquad \xi _{i,2}= \frac{-\beta _{i}-\sqrt{\beta _{i}^{2}+4\alpha _{i}}}{2}\quad (i=1,2), $$

and let \(G_{i,j}(t,s)\) (\(i,j=1,2\)) be the Green function of the linear boundary value problem

$$ \textstyle\begin{cases} -u_{i}''(t)+\xi _{i,j}u_{i}(t)=0, \quad 0< t< 1; \\ u_{i}(0)=u_{i}(1)=0, \quad i,j=1,2. \end{cases} $$

Then, for \(h_{i}\in C[0,1]\), the solution to the following linear boundary value problem

$$ \textstyle\begin{cases} u_{i}^{(4)}(t)+\beta _{i}u_{i}''(t)-\alpha _{i}u_{i}(t)=h_{i}(t), \quad 0< t< 1; \\ u_{i}(0)=u_{i}(1)=u_{i}''(0)=u_{i}''(1)=0, \quad i,j=1,2 \end{cases} $$

can be expressed as

$$ u_{i}(t)= \int _{0}^{1} \int _{0}^{1}G_{i,1}(t,\tau )G_{i,2}(\tau ,s)h_{i}(s)\,ds\,d \tau , \quad t\in [0,1]. $$
(2.1)

Lemma 2.1

The function \(G_{i,j}(t,s)\) (\(i=1,2\)) has the following properties:

  1. (1)

    \(G_{i,j}(t,s)=G_{i,j}(s,t)\) and \(G_{i,j}(t,s)>0\) for \(t,s\in (0,1)\);

  2. (2)

    \(G_{i,j}(t,s)\leq C_{i,j}G_{i,j}(s,s)\) for \(t,s\in [0,1]\), where \(C_{i,j}>0\) is a constant;

  3. (3)

    \(G_{i,j}(t,s)\geq \delta _{i,j}G_{i,j}(t,t)G_{i,j}(s,s)\) for \(t,s\in [0,1]\), where \(\delta _{i,j}>0\) is a constant;

  4. (4)

    \(G_{2,j}(t,s)\leq N_{j}G_{1,j}(t,s)\) for \(t,s\in [0,1]\), where \(N_{j}>0\) is a constant.

Proof

(1)–(3) can be seen from [9]. In addition, by careful calculation and Lemma 2.1 in [9], it is not difficult to prove that \(N_{j}:=\sup_{0< t,s<1}\frac{G_{2,j}(t,s)}{G_{1,j}(t,s)}<+ \infty \). Immediately, (4) is derived. □

The main tool used here is the following fixed point index theory.

Lemma 2.2

([6])

Let \(E_{1}\) be a Banach space and P be a cone in \(E_{1}\). Denote \(P_{r}=\{u\in P: \|u\|< r\}\) and \(\partial P_{r}=\{u\in P: \|u\|=r\}\) (\(\forall r>0\)). Let \(T:P\rightarrow P\) be a complete continuous mapping, then the following conclusions are valid.

  1. (1)

    If \(\mu Tu \neq u\) for \(u\in \partial P_{r} \) and \(\mu \in (0,1]\), then \(i(T,P_{r},P)=1\);

  2. (2)

    If \(\inf_{u\in \partial P_{r}}\|Tu\|>0\) and \(\mu Tu \neq u\) for \(u\in \partial P_{r}\) and \(\mu \geq 1\), then \(i(T,P_{r},P)=0\).

Conversion of boundary value problem (1.1)

In order to overcome the difficulties arising from singularity and semipositone, we convert boundary value problem (1.1) into another form (see (3.4)). For simplicity and convenience, set

$$ C_{1}= \int _{0}^{1}G_{1,1}(\tau ,\tau )G_{1,2}(\tau ,\tau )\,d\tau ,\qquad M_{i,j}= \max _{t\in [0,1]}G_{i,j}(t,t). $$

Then \(C_{1}\) and \(M_{i,j}\) (\(i, j=1,2\)) are positive numbers.

Now let us list the following assumptions which will be satisfied throughout the paper.

  1. (H1)

    There exist functions \(p\in L^{1}[J,\mathbb{R}^{+}]\) such that

    $$ f_{1}(t,u,v)+p(t)\geq 0, \quad \forall (t,u,v)\in J\times \mathbb{R}_{0}^{+} \times \mathbb{R}. $$
  2. (H2)

    \(f_{1}, f_{2}\in C[J\times \mathbb{R}_{0}^{+}\times \mathbb{R}, \mathbb{R}]\), and there exists \(N_{3}>0\) such that

    $$ \bigl\vert f_{2}(t,u,v) \bigr\vert \leq N_{3}\cdot \bigl[f_{1}(t,u,v)+p(t)\bigr]. $$

In this paper, the basic space is \(E:=C[0,1]\times C[0,1]\). It is a Banach space endowed with the norm \(\|(u,v)\|=\max \{N\|u\|,\|v\|\}\) for \((u,v)\in E\), where \(\|u\|=\max_{t\in [0,1]}|u(t)|\), \(\|v\|=\max_{t\in [0,1]}|v(t)|\), and \(N:=N_{1}N_{2}N_{3}\). \(N_{1}\), \(N_{2}\), and \(N_{3}\) are defined in Lemma 2.1 and (H2), respectively.

Moreover, let

$$\begin{aligned}& \mbox{\pounds} \bigl[u(t)\bigr]=u^{(4)}(t)+\beta _{1}u''(t)- \alpha _{1}u(t), \\& \Im \bigl[u(t)\bigr]=u^{(4)}(t)+\beta _{2}u''(t)- \alpha _{2}u(t), \\& r_{p}= \frac{[C_{1,1}C_{1,2}]^{2}M_{1,1}M_{1,2}}{\delta _{1,1}\delta _{1,2}C_{1}} \int _{0}^{1}p(s)\,ds. \end{aligned}$$
(3.1)

Define a function \(w: [0,1]\rightarrow \mathbb{R}^{+}\) by

$$ w(t)= \int _{0}^{1} \int _{0}^{1}G_{1,1}(t,\tau )G_{1,2}(\tau ,s)p(s)\,ds\,d \tau . $$

Applying (3.1) and Lemma 2.1, one can easily obtain that

$$ \begin{aligned} w(t)&= \int _{0}^{1} \int _{0}^{1}G_{1,1}(t,\tau )G_{1,2}( \tau ,s)p(s)\,ds\,d\tau \\ &\leq C_{1,1}C_{1,2}M_{1,2}G_{1,1}(t,t) \int _{0}^{1}p(s)\,ds=r_{p} \frac{\delta _{1,1}\delta _{1,2}C_{1}}{C_{1,1}C_{1,2}M_{1,1}}G_{1,1}(t,t),\quad t \in [0,1]. \end{aligned} $$
(3.2)

This together with (2.1) guarantees that \(w(t)\) is the positive solution of the following boundary value problem:

$$ \textstyle\begin{cases} \mbox{\pounds} [w(t)]=p(t), \quad 0< t< 1; \\ w(0)=w(1)=w''(0)=w''(1)=0. \end{cases} $$
(3.3)

Now we are in a position to convert SBVPs (1.1) into an approximate boundary value problem. For this matter, it will be carried out in two steps.

Firstly, in order to overcome the difficulties arising from semipositone, consider the following singular nonlinear differential boundary value problem:

$$ \textstyle\begin{cases} \mbox{\pounds} [u(t)]=f_{1}(t,[u(t)-w(t)]^{*},v(t))+p(t), \quad 0< t< 1; \\ \Im [v(t)]=f_{2}(t,[u(t)-w(t)]^{*},v(t)),\quad 0< t< 1; \\ u(0)=u(1)=u''(0)=u''(1)=0; \\ v(0)=v(1)=v''(0)=v''(1)=0, \end{cases} $$
(3.4)

where

$$ \bigl[u(t)\bigr]^{*} = \textstyle\begin{cases} u(t), &u(t)\geq 0; \\ 0 , &u(t)\leq 0. \end{cases} $$

Then we can obtain the following conclusion.

Lemma 3.1

Assume that \((u,v)\) is a solution of BVPs (3.4) and \(u(t)\geq w(t)\) for \(t\in [0,1]\). Then \((u,v)\) is a solution of SBVPs (1.1).

Proof

Let the vector \((u,v)\) be a solution of BVPs (3.4) and \(u(t)\geq w(t)\) for \(t\in [0,1]\). Then the definition of function \([\cdot ]^{*}\) together with (3.4) guarantees that

$$ \textstyle\begin{cases} \mbox{\pounds} [u(t)]=f_{1}(t,u(t)-w(t),v(t))+p(t),\quad 0< t< 1; \\ \Im [v(t)]=f_{2}(t,u(t)-w(t),v(t)), \quad 0< t< 1; \\ u(0)=u(1)=u''(0)=u''(1)=0; \\ v(0)=v(1)=v''(0)=v''(1)=0. \end{cases} $$
(3.5)

Set \(u_{1}=u-w\), \(v_{1}=v\). Then \(\mbox{\pounds} [u_{1}(t)]=\mbox{\pounds} [u(t)]-\mbox{\pounds} [w(t)]\) and \(\Im [v_{1}(t)]=\Im [v(t)]\), which implies

$$\begin{aligned}& \mbox{\pounds} \bigl[u(t)\bigr]=\mbox{\pounds} \bigl[u_{1}(t)\bigr]+\mbox{\pounds} \bigl[w(t)\bigr]=\mbox{\pounds} \bigl[u_{1}(t)\bigr]+p(t), \\& \Im \bigl[v(t)\bigr]=\Im \bigl[v_{1}(t)\bigr],\quad t\in [0,1]. \end{aligned}$$

So, (3.3) together with (3.5) guarantees that

$$ \textstyle\begin{cases} \mbox{\pounds} [u_{1}(t)]=f_{1}(t,u_{1}(t),v_{1}(t)),\quad 0< t< 1; \\ \Im [v_{1}(t)]=f_{2}(t,u_{1}(t),v_{1}(t)), \quad 0< t< 1; \\ u_{1}(0)=u_{1}(1)=u_{1}''(0)=u_{1}''(1)=0; \\ v_{1}(0)=v_{1}(1)=v_{1}''(0)=v_{1}''(1)=0. \end{cases} $$
(3.6)

Namely \((u_{1},v_{1})=(u-w,v)\) is a solution of SBVPs (1.1). □

Secondly, in order to overcome the singularity associated with SBVPs (1.1), consider the following approximate boundary value problem:

$$ \textstyle\begin{cases} \mbox{\pounds} [u(t)]=f_{1}^{j}(t,[u(t)-w(t)]_{j}^{*},v(t))+p(t), \quad 0< t< 1; \\ \Im [v(t)]=f_{2}^{j}(t,[u(t)-w(t)]_{j}^{*},v(t)), \quad 0< t< 1; \\ u(0)=u(1)=u''(0)=u''(1)=0; \\ v(0)=v(1)=v''(0)=v''(1)=0, \end{cases} $$
(3.7)

where

$$ f_{i}^{j}\bigl(t,[u]_{j}^{*},v \bigr) ] = \textstyle\begin{cases} f_{i}(t,u+\frac{1}{j},v),&u\geq 0; \\ f_{i}(t,\frac{1}{j},v),&u< 0 \ (i=1,2; j\in \mathbb{N}). \end{cases} $$

In the following, we shall mainly discuss the existence results for BVPs (3.7) by using the fixed point index theory. For this matter, first we define the following mappings:

$$\begin{aligned}& T_{1}^{j}(u,v) (t)= \int _{0}^{1} \int _{0}^{1}G_{1,1}(t,\tau )G_{1,2}( \tau ,s)\bigl[f_{1}^{j}\bigl(s, \bigl[u(s)-w(s)\bigr]^{*}_{j},v(s)\bigr)+p(s)\bigr]\,ds \,d\tau , \\& T_{2}^{j}(u,v) (t)= \int _{0}^{1} \int _{0}^{1}G_{2,1}(t,\tau )G_{2,2}( \tau ,s)f_{2}^{j}\bigl(s, \bigl[u(s)-w(s)\bigr]^{*}_{j},v(s)\bigr)\,ds\,d\tau , \\& T^{j}(u,v) (t)=\bigl(T_{1}^{j}(u,v) (t),T_{2}^{j}(u,v) (t)\bigr), \quad \forall t \in [0,1], (u,v) \in E, j\in \mathbb{N}. \end{aligned}$$
(3.8)

Obviously, it is easy to see that the existence of nontrivial solutions for BVPs (3.7) is equivalent to the existence of the nontrivial fixed point of \(T^{j}\). Therefore, we just need to find the nontrivial fixed point of \(T^{j}\) in the following work.

For the sake of obtaining the nontrivial fixed point of operator \(T^{j}\), set

$$ P=\bigl\{ (u,v)\in E: u(t)\geq \sigma (t) \Vert u \Vert \mbox{ and } \bigl\vert v(t) \bigr\vert \leq Nu(t), \forall t\in [0,1]\bigr\} , $$

where \(\sigma (t)=\frac{\delta _{1,1}\delta _{1,2}C_{1}}{C_{1,1}C_{1,2}M_{1,1}}G_{1,1}(t,t)\) and \(N=N_{1}N_{2}N_{3}\). \(N_{1}\), \(N_{2}\), and \(N_{3}\) are defined in Lemma 2.1 and (H2), respectively.

Evidently, P is a nonempty, convex, and closed subset of E. Furthermore, one can prove that P is a cone of Banach space E. For simplicity, denote

$$ P_{r}:=\bigl\{ (u,v)\in P: \bigl\Vert (u,v) \bigr\Vert < r\bigr\} . $$

Then, by the definition of cone P and the norm \(\|(u,v)\|\), one can see that

$$\begin{aligned}& \partial P_{r}:=\bigl\{ (u,v)\in P: \bigl\Vert (u,v) \bigr\Vert =r\bigr\} =\biggl\{ (u,v)\in P: \Vert u \Vert = \frac{r}{N}\biggr\} , \\& \overline{P_{r}}:=\bigl\{ (u,v)\in P: \bigl\Vert (u,v) \bigr\Vert \leq r\bigr\} =\biggl\{ (u,v)\in P: \Vert u \Vert \leq \frac{r}{N}\biggr\} . \end{aligned}$$

Clearly, for each \(r>0\), \(P_{r}\) is a relatively open and bounded set of P.

Main results

In this section, we present the main results of this paper. To do this, first we need to investigate the properties of mapping \(T^{j}\) (\(j\in \mathbb{N}\)).

Lemma 4.1

Assume that (H1) and (H2) hold. Then, for any \(j\in \mathbb{N}\), \(T^{j}:P\rightarrow P\) is completely continuous and \(T^{j}(P)\subset P\).

Proof

For \((u,v)\in P\), by virtue of Lemma 2.1, one can easily get that

$$\begin{aligned} T_{1}^{j}(u,v) (t)&= \int _{0}^{1} \int _{0}^{1}G_{1,1}(t,\tau )G_{1,2}( \tau ,s)\bigl[f_{1}^{j}\bigl(s, \bigl[u(s)-w(s)\bigr]^{*}_{j},v(s)\bigr)+p(s)\bigr]\,ds \,d\tau \\ &\geq \frac{\delta _{1,1}\delta _{1,2}C_{1}}{C_{1,1}C_{1,2}M_{1,1}}G_{1,1}(t,t) \bigl\Vert T_{1}^{j}(u,v) \bigr\Vert =\sigma (t) \bigl\Vert T_{1}^{j}(u,v) \bigr\Vert ,\quad \forall t\in [0,1], j \in \mathbb{N}. \end{aligned}$$

Moreover, (H2) together with Lemma 2.1 implies that

$$\begin{aligned} \bigl\vert T_{2}^{j}(u,v) (t) \bigr\vert &= \biggl\vert \int _{0}^{1} \int _{0}^{1}G_{2,1}(t,\tau )G_{2,2}( \tau ,s)f^{j}_{2}\bigl(s, \bigl[u(s)-w(s)\bigr]^{*}_{j},v(s)\bigr)\,ds\,d\tau \biggr\vert \\ &\leq N_{3} \int _{0}^{1} \int _{0}^{1}G_{2,1}(t,\tau )G_{2,2}(\tau ,s)f_{1}^{j}\bigl(s, \bigl[u(s)-w(s)\bigr]^{*}_{j},v(s)\bigr)\,ds\,d \tau \\ &\leq N_{1}N_{2}N_{3} \int _{0}^{1} \int _{0}^{1}G_{1,1}(t,\tau )G_{1,2}( \tau ,s)f_{1}^{j}\bigl(s, \bigl[u(s)-w(s)\bigr]^{*}_{j},v(s)\bigr)\,ds\,d\tau \\ &=N \bigl\vert T_{1}^{j}(u,v) (t) \bigr\vert ,\quad \forall t\in [0,1], j\in \mathbb{N}. \end{aligned}$$

Therefore, \(T^{j}(u,v)\in P\), namely \(T^{j}(P)\subset P\). In addition, notice that \(f_{1}\), \(f_{2}\), and \(G_{i,j}\) are continuous, one can deduce that \(T^{j}\) is completely continuous for each \(j\in \mathbb{N}\) by using normal methods such as Ascoli–Arzela theorem, etc. □

For convenience of expression, for each \(R_{1}>r_{1}>Nr_{p}\), take

$$ \Lambda _{[r_{1},R_{1}](t)}=\biggl(\biggl(\frac{r_{1}}{N}-r_{p} \biggr)\sigma (t), \frac{R_{1}}{N}+1\biggr]\times [-R_{1},R_{1}], $$

where \(r_{p}\) is defined in (3.1).

At the same time, define a functional \(\digamma :L^{1}(J)\rightarrow \mathbb{R}^{+}\) by

$$ \digamma (y)=\max_{t\in [0,1]} \int _{0}^{1} \int _{0}^{1}G_{1,1}(t, \tau )G_{1,2}(\tau ,s)y(s)\,ds\,d\tau ,\quad \text{for } y\in L^{1}(J).$$
(4.1)

Next, let us list the following assumptions which will be used in what follows.

(H1′):

For each \(R_{1}>r_{1}>Nr_{p}\), there exists \(\Psi _{r_{1},R_{1}}\in L^{1}(J)\) such that

$$ 0\leq f_{1}(t,u,v)+p(t)\leq \Psi _{r_{1},R_{1}}(t),\quad \text{for } \forall (t,u,v) \in J\times \Lambda _{[r_{1},R_{1}](t)}.$$
(4.2)
(H3):

There exist \(R>r>Nr_{p}\) and function \(\Phi _{r}\) such that

  1. (1)

    \(f_{1}(t,u,v)+p(t)\geq \Phi _{r}(t)\), \(\forall (t,u,v)\in J \times \Lambda _{[r,r](t)}\);

  2. (2)

    \(\digamma (\Psi _{R,R})<\frac{R}{N}\), \(\digamma (\Phi _{r})> \frac{r}{N}\).

Now we are in a position to give the following two lemmas to calculate the fixed point index of \(T^{j}\) (\(j\in \mathbb{N}\)) in \(P_{r}\).

Lemma 4.2

Assume that (H1′) and (H2)(H3) hold. Then the following conclusions are valid:

  1. (i)

    For any \(j\in \mathbb{N}\), \(i(T^{j}, P_{r},P)=0\);

  2. (ii)

    For any \(j\in \mathbb{N}\), \(i(T^{j},P_{R},P)=1\).

Proof

(i) For the sake of obtaining the desired result, we firstly prove that

$$ \inf_{(u,v)\in \partial P_{r}} \bigl\Vert T^{j}(u,v) \bigr\Vert >0\quad \mbox{and}\quad (u,v) \neq \mu T^{j}(u,v),\quad \forall (u,v)\in \partial P_{r}, \mu \geq 1 \mbox{ and } j\in \mathbb{N}. $$
(4.3)

In fact, if it is not true, then there exist \(\mu _{0}\geq 1\) and \((u_{0},v_{0})\in \partial P_{r}\) such that \((u_{0},v_{0})=\mu _{0} T^{j}(u_{0},v_{0})\). By (3.1), (3.2), and the definition of cone P, one can obtain that

$$ u_{0}(t)\geq \sigma (t) \Vert u_{0} \Vert = \frac{r}{N}\sigma (t), \qquad w(t)\leq r_{p} \sigma (t),\quad t\in [0,1]. $$

That is,

$$ u_{0}(t)-w(t)\geq \frac{r}{N}\sigma (t)-r_{p} \sigma (t)= \biggl(\frac{r}{N}-r_{p}\biggr) \sigma (t)\geq 0. $$

Moreover, by the definition of function \([\cdot ]^{*}_{j}\), we have

$$\begin{aligned}& \bigl[u_{0}(t)-w(t)\bigr]^{*}_{j}=u_{0}(t)-w(t)+ \frac{1}{j}\leq u_{0}(t)+ \frac{1}{j}\leq \frac{r}{N}+1, \\& \bigl\vert v_{0}(t) \bigr\vert \leq Nu_{0}(t)\leq r, \quad \forall t\in [0,1], j\in \mathbb{N}, \end{aligned}$$
(4.4)

which means

$$ \bigl(\bigl[u_{0}(t)-w(t)\bigr]^{*}_{j}, v_{0}(t)\bigr)\in \Lambda _{[r,r](t)},\quad \forall t \in [0,1], j \in \mathbb{N}. $$

Hence, applying \((u_{0},v_{0})=\mu _{0}T^{j}(u_{0},v_{0})\) and (H3), we obtain immediately that

$$ \begin{aligned} u_{0}(t)&= \mu _{0}T^{j}_{1}(u_{0},v_{0}) (t)\geq T^{j}_{1}(u_{0},v_{0}) (t) \\ &= \int _{0}^{1} \int _{0}^{1}G_{1,1}(t,\tau )G_{1,2}( \tau ,s)\bigl[f_{1}^{j}\bigl(t, \bigl[u_{0}(t)-w(t)\bigr]^{*}_{j},v_{0}(t) \bigr)+p(t)\bigr]\,ds\,d\tau \\ &\geq \int _{0}^{1} \int _{0}^{1}G_{1,1}(t,\tau )G_{1,2}( \tau ,s)\Phi _{r}(s)\,ds\,d\tau . \end{aligned} $$
(4.5)

Taking the maximum for both sides of (4.5) in \([0,1]\), we get

$$ \Vert u_{0} \Vert \geq \digamma (\Phi _{r})> \frac{r}{N}, \quad j\in \mathbb{N}. $$

This is in contradiction with \((u_{0},v_{0})\in \partial P_{r}\). Besides, it is clear that \(\inf_{(u,v)\in \partial P_{r}}\|T^{j}(u,v)\|>0\) by (4.5), and then (4.3) holds.

(ii) Next, we claim that

$$ (u,v)\neq \mu T^{j}(u,v),\quad \forall (u,v)\in \partial P_{R}, \mu \in (0,1], \mbox{and } j\in \mathbb{N}. $$
(4.6)

Suppose on the contrary that there exist \(\mu _{\lambda }\in (0,1]\) and \((u_{\lambda },v_{\lambda })\in \partial P_{R}\) such that \((u_{\lambda },v_{\lambda })=\mu _{\lambda } T^{j}(u_{\lambda },v_{ \lambda })\). Using a similar process of the proof as (i), we immediately get that

$$\begin{aligned}& \bigl\vert v_{\lambda }(t) \bigr\vert \leq Nu_{\lambda }(t)\leq R, \\& u_{\lambda }(t)-w(t)\geq \biggl(\frac{R}{N}-r_{p} \biggr)\sigma (t)\geq 0, \\& \bigl[u_{0}(t)-w(t)\bigr]^{*}_{j}=u_{0}(t)-w(t)+ \frac{1}{j}\leq u_{0}(t)+ \frac{1}{j}\leq \frac{R}{N}+1, \end{aligned}$$
(4.7)

which indicates

$$ \bigl(\bigl[u_{0}(t)-w(t)\bigr]^{*}_{j}, v_{0}(t)\bigr)\in \Lambda _{[R, R](t)}, \quad \forall t \in [0,1], j \in \mathbb{N}. $$

In addition, \((u_{\lambda },v_{\lambda })=\mu _{\lambda } T^{j}(u_{\lambda },v_{ \lambda })\) together with (4.2), (4.7), and (H3) deduces that

$$\begin{aligned} u_{\lambda }(t)&=\mu _{\lambda }T^{j}_{1}(u_{\lambda },v_{\lambda }) (t) \leq T^{j}_{1}(u_{\lambda },v_{\lambda }) (t) \\ &= \int _{0}^{1} \int _{0}^{1}G_{1,1}(t,\tau )G_{1,2}( \tau ,s)\bigl[f_{1}^{j}\bigl(s, \bigl[u_{\lambda }(s)-w(s)\bigr]^{*}_{j},v_{\lambda }(s) \bigr)+p(s)\bigr]\,ds\,d \tau \\ &\leq \int _{0}^{1} \int _{0}^{1}G_{1,1}(t,\tau )G_{1,2}( \tau ,s)\Psi _{R,R}(s)\,ds\,d\tau , \end{aligned}$$

which implies

$$ \Vert u_{\lambda } \Vert \leq \digamma (\Psi _{R,R})< \frac{R}{N},\quad j\in \mathbb{N}. $$

This is in contradiction with \((u_{\lambda },v_{\lambda })\in \partial P_{R}\). Therefore, (4.6) holds. To sum up, the proof is complete. □

Lemma 4.3

Assume that (H1′) and (H2)(H3) hold. In addition, suppose that:

  1. (H4)

    There exists an interval \([\alpha ,\beta ]\subset J\) such that

    $$ \lim_{\stackrel{|v|\leq Nu}{u\rightarrow +\infty }} \min_{t\in [\alpha ,\beta ]} \frac{f_{1}(t,u,v)}{u}=+\infty . $$

Then there exists a constant \(R^{*}>R\) such that \(i(T^{j},P_{R^{*}},P)=0\) for each \(j\in \mathbb{N}\).

Proof

First, choose a positive number ϒ satisfying

$$ \Upsilon >2 \biggl(\min_{t\in [\alpha ,\beta ]}\sigma (t) \cdot \max_{t\in [0,1]} \int _{\alpha }^{\beta } \int _{\alpha }^{\beta }G_{1,1}(t,\tau )G_{1,2}(\tau ,s)\,ds\,d\tau \biggr)^{-1}. $$
(4.8)

Then, by (H4), it is easy to see that there exists \(\ell >\frac{R}{N}\) such that

$$ \frac{f_{1}(t,u,v)}{u}\geq \Upsilon ,\quad \forall t\in [\alpha , \beta ], u \geq \ell , \vert v \vert \leq Nu. $$
(4.9)

Let \(R^{*}\) be a positive number satisfying \(R^{*}>\frac{2N\ell }{\min_{t\in [\alpha ,\beta ]}\sigma (t)}\). Then

$$ \frac{2R}{N}< 2\ell < \frac{R^{*}}{N}. $$
(4.10)

Next we show

$$ (u,v)\neq \mu T^{j}(u,v),\quad \forall (u,v)\in \partial P_{R^{*}}, \mu \geq 1, \mbox{and } j\in \mathbb{N}. $$
(4.11)

In fact, if it is not true, then there exist \(\mu _{0}\geq 1\) and \((u_{0},v_{0})\in \partial P_{R^{*}}\) such that \((u_{0},v_{0})=\mu _{0} T^{j}(u_{0},v_{0})\). Therefore, for any \(t\in [\alpha ,\beta ]\), by (3.1), (3.2), and (4.10), one can easily get that

$$ \begin{aligned} u_{0}(t)-w(t)&\geq u_{0}(t)-r_{p}\sigma (t)\geq u_{0}(t)- \frac{R}{N}\sigma (t) \\ & \geq u_{0}(t)-\frac{R}{N}\cdot \frac{N}{R^{*}}u_{0}(t)=u_{0}(t)- \frac{R}{R^{*}}u_{0}(t) \\ & \geq \frac{u_{0}(t)}{2} \geq \frac{R^{*}}{2 N}\cdot \min _{t\in [\alpha ,\beta ]}\sigma (t)> \ell >0. \end{aligned} $$
(4.12)

Hence, from (4.9) and (4.12), we have

$$ \begin{aligned} u_{0}(t)&= \mu _{0} T_{1}^{j}(u_{0},v_{0}) (t)\geq T_{1}^{j}(u_{0},v_{0}) (t) \\ &= \int _{0}^{1} \int _{0}^{1}G_{1,1}(t,\tau )G_{1,2}( \tau ,s)\bigl[f_{1}^{j}\bigl(t, \bigl[u_{0}(s)-w(s)\bigr]^{*}_{j},v_{0}(s) \bigr)+p(s)\bigr]\,ds\,d\tau \\ &= \int _{0}^{1} \int _{0}^{1}G_{1,1}(t,\tau )G_{1,2}( \tau ,s)\biggl[f_{1}\biggl(t,u_{0}(s)-w(s)+ \frac{1}{j},v_{0}(s)\biggr)+p(s)\biggr]\,ds\,d\tau \\ &\geq \int _{0}^{1} \int _{0}^{1}G_{1,1}(t,\tau )G_{1,2}( \tau ,s)\cdot f_{1}\biggl(t,u_{0}(s)-w(s)+ \frac{1}{j},v_{0}(s)\biggr)\,ds\,d\tau \\ &\geq \Upsilon \cdot \int _{\alpha }^{\beta } \int _{ \alpha }^{\beta }G_{1,1}(t,\tau )G_{1,2}(\tau ,s)\cdot \biggl(u_{0}(s)-w(s)+ \frac{1}{j}\biggr)\,ds\,d\tau \\ &\geq \frac{\Upsilon R^{*}}{2N}\min_{t\in [ \alpha ,\beta ]}\sigma (t) \int _{\alpha }^{\beta } \int _{\alpha }^{\beta }G_{1,1}(t, \tau )G_{1,2}(\tau ,s)\,ds\,d\tau . \end{aligned} $$
(4.13)

Consequently, by (4.8) and (4.13), we immediately obtain that

$$ \Vert u_{0} \Vert \geq \frac{\Upsilon R^{*}}{2N}\min _{t \in [\alpha ,\beta ]}\sigma (t)\cdot \biggl(\max_{t\in [\alpha , \beta ]} \int _{\alpha }^{\beta } \int _{\alpha }^{\beta }G_{1,1}(t,\tau )G_{1,2}( \tau ,s)\,ds\,d\tau \biggr)>\frac{R^{*}}{N}. $$

This is in contradiction with \((u_{0},v_{0})\in \partial P_{R^{*}}\). Moreover, in view of (4.13) we know that \(\inf_{(u,v)\in \partial P_{R^{*}}}\|T^{j}(u,v)\|>0\). So, by Lemma 2.2, the conclusion of this lemma follows. □

Now, we are in a position to prove the main theorem of the present paper.

Theorem 4.4

Under assumptions (H1′) and (H2)(H4), SBVPs (1.1) admits at least two nontrivial solutions.

Proof

This proof will be carried out in four steps.

Claim 1

System (3.7) has at least two nontrivial solutions.

In fact, applying Lemmas 4.24.3 and the additivity of the fixed point index, one can get for any \(j\in \mathbb{N}\) that

$$\begin{aligned}& i\bigl(T^{j},P_{R^{*}}\setminus \overline{ P_{R}},P\bigr)=i\bigl(T^{j},P_{R^{*}},P \bigr)-i\bigl(T^{j}-P_{R},P\bigr)=0-1=-1, \\& i\bigl(T^{j},P_{R}\setminus \overline{ P_{r}},P\bigr)=i\bigl(T^{j},P_{R},P \bigr)-i\bigl(T^{j}-P_{r},P\bigr)=1-0=1. \end{aligned}$$

So, there exist \((u_{j},v_{j})\in P_{R^{*}}\setminus \overline{ P_{R}}\) and \((U_{j},V_{j})\in P_{R}\setminus \overline{ P_{r}}\) satisfying

$$ (u_{j},v_{j})=T^{j}(u_{j},v_{j}) \quad \mbox{and} \quad (U_{j},V_{j})=T^{j}(U_{j},V_{j}),\quad j \in \mathbb{N}. $$

Namely, system (3.7) has at least two nontrivial solutions satisfying

$$ r< N \Vert u_{j} \Vert < R< N \Vert U_{j} \Vert < R^{*}. $$
(4.14)

Claim 2

\(\{(u_{j},v_{j})\}_{j\in \mathbb{N}}\) and \(\{(U_{j},V_{j})\}_{j\in \mathbb{N}}\) are bounded equicontinuous families on \([0,1]\).

Notice that the boundedness is obvious. To prove the equicontinuity, let us prove that \(\{u_{j}\}_{j\in \mathbb{N}}\) are equicontinuous on \([0,1]\) first. Since

$$ u_{j}(t)-w(t)\geq \biggl(\frac{r}{N}-r_{p} \biggr)\sigma (t)\geq 0, $$
(4.15)

applying (4.2) and (4.15), we get that, for any \(0< t_{1}< t_{2}<1\) and \(j\in \mathbb{N}\),

$$ \begin{aligned} \bigl\vert u_{j}(t_{1})-u_{j}(t_{2}) \bigr\vert &= \bigl\vert T_{1}^{j}(u_{j},v_{j}) (t_{1})-T_{1}^{j}(u_{j},v_{j}) (t_{2}) \bigr\vert \\ &= \int _{0}^{1} \int _{0}^{1} \bigl\vert G_{1,1}(t_{1}, \tau )-G_{1,1}(t_{2}, \tau ) \bigr\vert \cdot G_{1,2}(\tau ,s) \\ &\quad {}\cdot \bigl[f_{1}\bigl(s, \bigl[u_{j}(s)-w(s)\bigr]^{*}_{j},v_{j}(s) \bigr)+p(s)\bigr]\,ds\,d \tau \\ &\leq C_{1,2}\max_{t\in [0,1]}G_{1,2}(t,t) \int _{0}^{1} \int _{0}^{1}\bigl|G_{1,1}(t_{1}, \tau )-G_{1,1}(t_{2},\tau ) \bigr| \\ &\quad {}\cdot\Psi _{r,R^{*}}(s)\,ds \,d\tau \\ &\leq C_{1,2}\max_{t\in [0,1]}G_{1,2}(t,t) \cdot \int _{0}^{1}\Psi _{r,R^{*}}(s)\,ds \\ &\quad {}\cdot \int _{0}^{1} \bigl\vert G_{1,1}(t_{1}, \tau )-G_{1,1}(t_{2},\tau ) \bigr\vert \,d\tau . \end{aligned} $$
(4.16)

So, by (H1′), (4.16), and the continuity of \(G_{1,1}\), one can easily see that the equicontinuity of \(\{u_{j}\}_{j\in \mathbb{N}}\) holds. From a process similar to the above, we get that the equicontinuity of \(\{v_{j}\}_{j\in \mathbb{N}}\) holds by applying condition (H2). Therefore, \(\{(u_{j},v_{j})\}_{j\in \mathbb{N}}\) is an equicontinuous family on \(t\in [0,1]\). Very similarly, \(\{(U_{j},V_{j})\}_{j\in \mathbb{N}}\) is also an equicontinuous family on \([0,1]\).

To sum up, \(\{(u_{j},v_{j})\}_{j\in \mathbb{N}}\) and \(\{(U_{j},V_{j})\}_{j\in \mathbb{N}}\) are the bounded equicontinuous families on \([0,1]\). By the Arzela–Ascoli theorem, there exist subsequences of them such that

$$\begin{aligned}& (u_{j_{n}},v_{j_{n}})\rightarrow (u_{0},v_{0})\quad \text{as } n \rightarrow +\infty \text{ in } E, \\& (U_{j_{n}},V_{j_{n}})\rightarrow (U_{0},V_{0})\quad \text{as } n \rightarrow +\infty \text{ in } E. \end{aligned}$$
(4.17)

Claim 3

\((u_{0},v_{0})\) and \((U_{0},V_{0})\) are nontrivial solutions of BVPs (3.4).

Since \((u_{j_{n}},v_{j_{n}})\) satisfies the integral equations

$$ \begin{aligned} &u_{j_{n}}(t)= \int _{0}^{1} \int _{0}^{1}G_{1,1}(t,\tau )G_{1,2}( \tau ,s)\bigl[f_{1}^{j}\bigl(s, \bigl[u_{j_{n}}(s)-w(s)\bigr]^{*}_{j},v_{j_{n}}(s) \bigr)+p(s)\bigr]\,ds\,d \tau , \\ &v_{j_{n}}(t)= \int _{0}^{1} \int _{0}^{1}G_{2,1}(t,\tau )G_{2,2}( \tau ,s)f_{2}^{j}\bigl(s, \bigl[u_{j_{n}}(s)-w(s)\bigr]^{*}_{j},v_{j_{n}}(s) \bigr)\,ds\,d\tau . \end{aligned} $$
(4.18)

From (H1′) and the well-known Lebesgue dominated convergence theorem, one can get that

$$ \begin{aligned} &u_{0}(t)= \int _{0}^{1} \int _{0}^{1}G_{1,1}(t,\tau )G_{1,2}( \tau ,s)\bigl[f_{1}\bigl(s,\bigl[u_{0}(s)-w(s) \bigr]^{*},v_{0}(s)\bigr)+p(s)\bigr]\,ds\,d\tau , \\ &v_{0}(t)= \int _{0}^{1} \int _{0}^{1}G_{2,1}(t,\tau )G_{2,2}( \tau ,s)f_{2}\bigl(s,\bigl[u_{0}(s)-w(s) \bigr]^{*},v_{0}(s)\bigr)\,ds\,d\tau . \end{aligned} $$
(4.19)

Therefore, \((u_{0},v_{0})\) is a nontrivial solution of BVPs (3.4). Similarly, we also get that \((U_{0},V_{0})\) is a nontrivial solution of BVPs (3.4). In addition, it is obvious that \(u_{0}(t)-w(t)\geq 0\) and \(U_{0}(t)-w(t)\geq 0\). Then, from Lemma 3.1, we know that \((u_{0}-w(t),v_{0})\) and \((U_{0}(t)-w(t),V_{0}(t))\) are the nontrivial solution of SBVPs (1.1).

Claim 4

\((u_{0},v_{0})\neq (U_{0},V_{0})\).

Since \((u_{0},v_{0})\subset P_{R^{*}}\setminus \overline{P_{R}}\) and \((U_{0},V_{0})\subset P_{R}\setminus \overline{P_{r}}\), we only need to prove that BVPs (3.4) has no solutions on \(\partial P_{R}\). Suppose on the contrary that there exists \((\tilde{u},\tilde{v})\in \partial P_{R}\) satisfying BVPs (3.4). Then

$$ \begin{aligned} \tilde{u}(t)&= \int _{0}^{1} \int _{0}^{1}G_{1,1}(t,\tau )G_{1,2}( \tau ,s)\bigl[f_{1}\bigl(t,\bigl[ \tilde{u}(t)-w(t)\bigr]^{*},\tilde{v}(t)\bigr)+p(t)\bigr]\,ds\,d\tau \\ &\leq \int _{0}^{1} \int _{0}^{1}G_{1,1}(t,\tau )G_{1,2}( \tau ,s)\Psi _{R,R}(s)\,ds\,d\tau . \end{aligned} $$
(4.20)

Taking the maximum on both sides of (4.20) in \([0,1]\), one can easily obtain that

$$ \Vert \tilde{u} \Vert \leq \digamma \bigl(\Psi _{R,R}(s)\bigr)< \frac{R}{N}. $$

This is in contradiction with \((\tilde{u},\tilde{v})\in \partial P_{R}\). To sum up, the conclusion of this theorem follows. □

An example

In this section, an illustrative example is worked out to show the effectiveness of the obtained result.

Example 5.1

Consider the following boundary value problem of fourth-order differential systems:

$$ \textstyle\begin{cases} u^{(4)}(t)+u''(t)-\pi ^{2}u(t)=\frac{v}{\sqrt{t(1-t)}}(u^{2}+ \frac{1}{u})-\kappa \cos (\frac{\pi t}{2}), \quad 0< t< 1; \\ v^{(4)}(t)+\frac{1}{2}v''(t)-\frac{\pi ^{2}}{2}v(t)= \frac{v\cos (t)}{N_{1}N_{2}\sqrt{t(1-t)}}(u^{2}+\frac{\sin (t)}{u}), \quad 0< t< 1; \\ u(0)=u(1)=u''(0)=u''(1)=0; \\ v(0)=v(1)=v''(0)=v''(1)=0, \end{cases} $$
(5.1)

where \(\kappa = \frac{2[C_{1,1}C_{1,2}]^{2}M_{1,1}M_{1,2}}{100\pi \delta _{1,1}\delta _{1,2}C_{1}}\), \(u\in \mathbb{R}_{0}^{+}\), and \(\frac{1}{2}\leq |v|\leq 1\).

Conclusion: SBVPs (5.1) has at least two nontrivial solutions.

Proof

SBVPs (5.1) can be regarded as the form of SBVPs (1.1), where

$$\begin{aligned}& \alpha _{1}=\pi ^{2}, \qquad \beta _{1}=1,\qquad \alpha _{2}=\frac{\pi ^{2}}{2}, \qquad \beta _{2}= \frac{1}{2}, \\& f_{1}(t,u,v)=\frac{v}{\sqrt{t(1-t)}}\biggl(u^{2}+ \frac{1}{u}\biggr)-\kappa \cos (t), \end{aligned}$$

and

$$ f_{2}(t,u,v)=\frac{v\cos (t)}{N_{1}N_{2}\sqrt{t(1-t)}}\biggl(u^{2}+ \frac{\sin (t)}{u}\biggr). $$

Then

$$ \xi _{1,1}=\frac{-1+\sqrt{1+4\pi ^{2}}}{2},\qquad \xi _{1,2}= \frac{-1-\sqrt{1+4\pi ^{2}}}{2}. $$

Clearly, \(\alpha _{i}\) and \(\beta _{i}\) (\(i = 1,2\)) satisfy condition (1.2). Moreover, by careful calculation and Lemma 2.1 in [9], one can obtain that

$$\begin{aligned}& G_{1,1}(t,s) = \textstyle\begin{cases} \frac{\sinh w_{1,1}t\sinh w_{1,1}(1-s)}{w_{1,1}\sinh w_{1,1}}, &0\leq t \leq s\leq 1; \\ \frac{\sinh w_{1,1}s\sinh w_{1,1}(1-t)}{w_{1,1}\sinh w_{1,1}}, &0\leq s \leq t\leq 1, \end{cases}\displaystyle \\& G_{1,2}(t,s) = \textstyle\begin{cases} \frac{\sin w_{1,2}t\sin w_{1,2}(1-s)}{w_{1,2}\sin w_{1,2}}, &0\leq t \leq s\leq 1; \\ \frac{\sin w_{1,2}s\sin w_{1,2}(1-t)}{w_{1,2}\sin w_{1,2}}, &0\leq s \leq t\leq 1, \end{cases}\displaystyle \end{aligned}$$

where \(w_{1,i}=\sqrt{|\xi _{1,i}|}\) (\(i=1,2\)).

Take \(p(t)=\kappa \cos (\frac{\pi t}{2})\), and simple calculation implies that (H1′) holds. Moreover, (H2) holds by choosing \(N_{3}=\frac{1}{N_{1}N_{2}}\). For convenience, let

$$\begin{aligned}& \Delta _{1}=: 2C_{1,1}C_{1,2} \int _{0}^{1}G_{1,1}(\tau ,\tau )\,d \tau \int _{0}^{1}G_{1,2}(s,s)\,ds, \\& \Delta _{2}=: \delta _{1,1}\delta _{1,2}C_{1} \max_{t\in [0,1]}G_{1,1}(t,t) \int _{0}^{1}G_{1,2}(s,s)\,ds. \end{aligned}$$

Obviously, it is easy to get that \(r_{p}=\frac{1}{100}\) from (3.1) and \(N=1\). Moreover, choose

$$ r=\frac{ \sqrt{1+4\Delta _{2}}+1}{4}>Nr_{p}= \frac{1}{100},\qquad \Phi _{r}(t)= \frac{1}{2(r+1) \sqrt{t(1-t)}}, $$

and

$$ R>\max \bigl\{ (\Delta _{1})^{\frac{3}{2}},Nr\bigr\} , \qquad \Psi _{R,R}(t)= \frac{2(R+1)^{2}}{ \sqrt{t(1-t)}}. $$

Then careful calculation indicates that (H3) holds. From \([\alpha ,\beta ]\subset (0, 1)\), it follows that

$$ \lim_{\stackrel{|v|\leq Nu}{u\rightarrow +\infty }} \min_{t\in [\alpha ,\beta ]} \frac{f_{1}(t,u,v)}{u}=\lim_{ \stackrel{|v|\leq Nu}{u\rightarrow +\infty }} \min_{t\in [ \alpha ,\beta ]} \frac{ \frac{v}{ \sqrt{t(1-t)}}(u^{2}+\frac{1}{u}))-\kappa \cos ( \frac{\pi t}{2})}{u}=+ \infty , $$

which implies that condition (H4) holds. Consequently, SBVPs (5.1) has at least two nontrivial solutions by Theorem 4.4. □

Availability of data and materials

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Acknowledgements

The authors are thankful to the editor and anonymous referees for their valuable comments and suggestions.

Funding

Y. Liu was supported by NNSF of P.R. China (62073202) and the Natural Science Foundation of Shandong Province (ZR2020MA007). D. Zhao was supported by a project of Shandong Province Higher Educational Science and Technology Program of China under the grant J18KA233.

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All authors conceived of the study, participated in its design and coordination, drafted the manuscript, participated in the sequence alignment, and read and approved the final manuscript.

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Correspondence to Yansheng Liu.

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Lin, L., Liu, Y. & Zhao, D. Multiple solutions for singular semipositone boundary value problems of fourth-order differential systems with parameters. Bound Value Probl 2021, 79 (2021). https://doi.org/10.1186/s13661-021-01554-1

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Keywords

  • Multiple solutions
  • Singular semipositone problems
  • Cone
  • Fixed point index